SOLUTION: Mrs. Flores recently subscribed in a tv cable plan at GSAT Company. The receiving dish of the GSAT Cable is in the shape of a paraboloid of revolution. Find the location of the r

Question 1186192: Mrs. Flores recently subscribed in a tv cable plan at GSAT Company. The
receiving dish of the GSAT Cable is in the shape of a paraboloid of revolution.
Find the location of the receiver which is placed at the focus if the dish is 12
feet across and 3 feet deep. Also write the standard form equation of the ellipse.

Found 2 solutions by CPhill, ikleyn:
Answer by CPhill(1959)   (Show Source): You can put this solution on YOUR website!
Here's how to find the location of the receiver and the equation of the paraboloid:
**1. Set up a coordinate system:**
It's easiest to place the vertex of the paraboloid at the origin (0,0) and have the parabola open upwards. This makes the axis of symmetry along the y-axis.
**2. Identify key points:**
* The dish is 12 feet across, so it extends 6 feet to either side of the y-axis. This gives us two points on the parabola: (-6, 3) and (6, 3).
* The dish is 3 feet deep, so the vertex is at (0,0) and the focus is somewhere along the positive y-axis.
**3. Standard equation of a parabola:**
The standard form equation of a parabola opening upwards with its vertex at the origin is:
x² = 4py
where 'p' is the distance from the vertex to the focus.
**4. Solve for 'p' (the distance to the focus):**
We can use one of the points we identified, such as (6, 3), and plug it into the equation to solve for 'p':
6² = 4p * 3
36 = 12p
p = 3
**5. Location of the receiver (focus):**
Since the vertex is at (0,0) and p = 3, the focus is located at (0, 3). This means the receiver should be placed 3 feet above the vertex of the dish.
**6. Standard form equation of the paraboloid:**
Now that we know p = 3, we can plug it into the standard equation:
x² = 4 * 3 * y
x² = 12y
Therefore, the location of the receiver (focus) is **3 feet above the vertex**, and the standard form equation of the paraboloid is **x² = 12y**.

Answer by ikleyn(52802)   (Show Source): You can put this solution on YOUR website!
.
Mrs. Flores recently subscribed in a tv cable plan at GSAT Company.
The receiving dish of the GSAT Cable is in the shape of a paraboloid of revolution.
Find the location of the receiver which is placed at the focus if the dish is 12
feet across and 3 feet deep.
Also write the standard form equation of the ellipse.
~~~~~~~~~~~~~~~~~~~~~~~~~~

This post requests to "write the standard form equation for the ellipse",

                but the ellipse shape is irrelevant to the problem.

It is why I decided do not touch it, in order for "don't get my hands dirty",
and so this task has been hanging here for several months (if not years).

For other similar problem, which I solved at this forum recently, see the link

https://www.algebra.com/algebra/homework/Finance/Finance.faq.question.1186320.html

https://www.algebra.com/algebra/homework/Finance/Finance.faq.question.1186320.html

By the way, one more notice to the solution by @CPhill, to its last sentence:

            The standard form equation for this paraboloid is

                                + = 12z.

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